small-scale robotic arm
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Small-Scale Robotic Arm
Senior Capstone Project
Ben Boyle and Kitera Hayes
Project Advisor: Dr. Gary Dempsey
April 29, 2004
Outline Objectives Equipment List System Specifications Functional Description Block Diagram System Parameters System Identification Implementation of Controllers Flexible Rotary Joint System Limitations Conclusion Completed Tasks Questions
Objectives
Determination of Plant ModelFast System ResponseWide Command Range (± 90 degrees)High Stability Margin (GM, PM)User-friendly Software InterfaceLow Resonant Frequency Mode with New Arm
Equipment List
200 MHz Pentium-based computerQuanser System
Robotic Arm with Flexible Rotary Joint Power Amplifier
Software MATLAB (SIMULINK) Borland C
Lab Workstation
Robotic Arm
System Specifications
Command: ± 90 set points, ± 40 deg/sec velocityPercent Overshoot = 0 %Steady-State Error = ± 2 degrees Phase Margin 70 degrees
Functional Description
Positioning
Figure 1 - Input/Output Description
Command Input
Small Scale Robotic Arm
Control
Functional Description
Software InterfacePositioning
Modes of Operation
Block Diagram
System (Plant)Software
Figure 2 - Block Diagram of Robotic Arm
System Parameters
System (Plant) Amplifier 5 [V] @ 3 [A] Position Sensor 180 of travel DC motor 5 [V] External Gears 5:1 velocity reduction Internal Gears 14.1:1 velocity reduction Antialiasing Filter first-order low-pass with pole @ 163 [rad/sec]
Software 200 [MHz] PC A/D converter 12 bit plus sign, 5 [V] D/A converter 12 bit, 5 [V]
System Identification
Closed-loop ResultsOpen-loop ResultsPlant Model Equation Plant Model Verification
System Identification
Closed-loop Results Gain k = 0.025 Best Fit
Close to 0% overshoot
Step input of ±20° DC Gain
Gp(0) = 27°/[V]
System Identification
k=0.025 D/A GpR=20
E=12
Controller voltage=0.2954
C=8
Controller Voltage = (12°)(.025) = 0.295 [V]
DC Gain [Gp(0)] = 8°/0.295 [V] = 27°/[V]
Figure 3 – DC Gain Calculation of System
System Identification
Figure 4 - Gain k = 0.025, Step input of ±20°, Closed-loop
(Experimental Results)
System Identification
Open-loop Results Verify DC gain of plant Calculate accurate time delay Help to determine plant model
System Identification
Figure 5 - k = 1.0, Step input voltage of 0.74 [V], Open-loop
(Experimental Results)
System Identification
Input Voltage = 20°/(27°/[V])
= 0.74 [V] (Open-loop)
Command Degree Calculation:
(K)(Command Voltage)(DC Gain) = Command Degrees
Theoretical Command Degrees 20°
Experimental Command Degrees 17°
Percent Error = 17.6%
System Identification
Plant Gp = k[a/(s+a)2]c(t) = k[1-e-at - at(e-at)]
@ k = 1.0 and t = 2.86 seconds, c = 11.352° Double Pole @ a = -0.76
Pole Identification using Laplace Transform
System Identification
TypicalOpen-loop
Poles
Figure 6 – Second Order System (Poles = -0.76)
ActualOpen-loop
Double Pole
-0.76
System Identification
Plant Model Equation:
27e-0.0562s
(s/0.76 + 1) 2
(OPEN-LOOP)
System Identification
20.48º
Figure 7 - SIMULINK Scope Output for Open-loop System = 20.48º
Plant Model Verification
System Identification
8.38º
Figure 8 - SIMULINK Scope Output for Closed-loop System = 8.38º
Plant Model Verification
P Controller
Figure 9 - Theoretical P Controller Output Figure 10 - P Controller System Output
PI Controller
Figure 12 - PI Controller System OutputFigure 11 - Theoretical PI Controller Output
PID Controller
Figure 13 - Theoretical PID Controller Output Figure 14 - PID Controller System Output
Feed-Forward/PI Controller
Figure 15 - Feed-Forward/PI Controller Block Diagram
Feed-Forward/PI Controller
Figure 16 - Theoretical FF/PI Controller Output Figure 17 - FF/PI Controller System Output
Controller Comparison
P Controller FF/PI Controller
Figure 19 - FF/PI Controller System OutputFigure 18 - P Controller System Output
Flexible Rotary Joint
Flexible Rotary Joint
Figure 20 - P Controller System Output Figure 21 - P Controller Flex Joint System Output
System Limitations
D/A Converter ± 5 [V] Static Friction
Just matches the applied force to try and prevent motion
Modeling Time delay e-std (linear) Kinetic Friction
Moving friction with respect to speeds Inertia
J = (mass)(radius2) Gravity
System Limitations
(a) With Friction (b) Without Friction
Figure 22(a-b) – Friction Characteristics for Pendulum System
-B/2J
PENDULUM
System Limitations
Figure 23 - Closed-loop Time Delay and % Overshoot Calculations for Varying Gain k
Tdavg = 56.2 [ms]
Time Delay Gain k Percent Overshoot Time Delay (Td) 0.01 0.64 % 80 ms 0.015 0.76 % 66 ms 0.02 0.91 % 80 ms 0.025 2.29 % 57 ms 0.03 10.45 % 58 ms 0.035 8.07 % 19 ms 0.04 28.00 % 50 ms 0.045 26.70 % 47 ms 0.05 33.48 % 48 ms
Conclusion
PI Controller is slowPID Controller does not workSolution is FF/PI Controller
Completed Tasks
Plant Model and Validation Proportional, PI, and PID Controllers FF Controller with PI
User-friendly Software Interface
Future Work Plant Model for Flexible Rotary Joint Gripper Motor with Varying Loads Notch Filter Incorporation
Questions?
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